Slew (not as in “killed”), bother, glom, down (the direction), slogan, pet (noun), smashing, twig…there is suggestion that “slugger” and “jazz” came from the Irish, too. Any more? There seems to be a rich array of this stuff.
“Galore” (go leor) is an obvious one.
My copy of Slanguage: A Dictionary of Irish Slang by Bernard Share has a lot of these. Borrowings galore, in fact.
Such blather. 
I want to believe that Frank Zappa’s album title Chunga’s Revenge was in fact intentionally touching on this very subject. Nyuk.
I’m dubious, but a lot of people think “snazzy” comes from “snasail” (neat, elegant, ornamented).
Smithereens
Pixilated, pixel, pixie, etc.
Galore made it into english because of Compton Mackenzie’s novel Whisky Galore which was filmed under that name in the UK, but renamed Tight Little Island for the US market, IIRC.
It was suggested by one of my Oideas Gael teachers that the British slang “Smashing” meaning something very good comes from “is maith sin”.
A couple of others:
“Dig” (as in “you dig?”) came into the jazz lexicon via the Irish “an dtuigeann tú?” (“do you understand?”…also used as a general place holder, in the same way we might say “you know?”)
“Phoney.” Actually came from the Irish “fáinne” (ring). Evidently there was quite a lot of interest in the Claddagh ring among non-Irish in the late 19th and early 20th century, which led to a lot of cheap knock-offs. Somehow the Irish word came to be associated with the cheapies.
Dare we forget “whiskey”? Came into English via “uisce beatha.” Interestingly, got itself assimilated BACK into Irish as “fuisce”!
Redwolf
It has been suggested that the “An dtuigeann tú/You dig?” loan came from when you had large numbers of Irish-speaking laborers mixing with African American laborers when laying track for the great 19th century railroads…
One (rather dubious) example I heard over the summer was “moccasin”, which a Scottish friend suggested to me came from the Scottish Gaelic “mo chasan” (“my feet”). The story goes that a Scottish fur trapper was bartering with some Native Americans one day and, pointing to his worn out shoes, said something to the effect of “Have you got anything for ‘mo chasan?’”, to which he was given some leather slippers. Again, dubious to say the least, I think…
A bit off topic, but I also find it interesting the sorts of grammatical constructions that have entered into English (particularly North American dialects of English) via Irish or Gaelic. For instance, many of us probably remember Yousemite Sam’s signature line “That’ll learn him!” from the old Warner Brothers cartoons. Some of us may have rolled our eyes at older relatives that have used similar constructions. I have a feeling this probably originally came from Scottish Gaelic, where it would be a very natural and correct grammatical construction. Gaelic does have a verb meaning “to teach” (“teagaisg”), however, it’s only become widely used relatively recently. You still often hear older speakers use the construction “ag ionnsachadh + do”, e.g.
Tha mi ag ionnsachadh Gàidhlig dhut. (“I am learning Gaelic to you.”)
What are some other examples you can think of?
Gu dearbh. Moccasin come from the Algonquian languages. In Western Abenaki, for example, the root of the word for shoe is kezen, pronounced something like “guzzun”. But the rules of Abenaki grammar dictate that a noun has to belong to someone or something (if I’m remembering this right - the class I took was a long time ago), and so the shoe would have to be my shoe (n’kezen), your shoe (k’kezen), his/her shoe (w’kezen), etc. When there is no identifiable owner, you use the “fourth person” pronoun (aka third person obviative), i.e. “someone’s shoe”, which you’ve probably guessed by now is m’kezen.
And that’s more Eastern Algonquian grammar than anyone should have to see on C&F.
I can’t think of any grammatical influences on standard North American Englishes, but in places like Cape Breton you hear quite a few constructions that must have either come from Scottish Gaelic directly, or from Irish via Newfoundland (depending on whether they’re heard more in the country or in the towns). I’m thinking of things like “Come here till I see ya” and “I’m after having a cup of tea” (meaning I’ve already had it).
The word “bog” actually comes from the Irish adjective “bog,” which means “soft” (most likely by way of “talamh bog” – “soft ground.”) The actual Irish word for a bog (as in where you did turf) is “portach.” Of course, from that you get “bogged down” and other such terms meaning “stuck.”
The word “brogue” (as a slang term for an Irish accent) comes from the Irish word “bróg” (“shoe”). I’ve heard a couple of theories for this, both derogatory: The first being that it started with the English making fun of the fact that the Irish often didn’t have shoes to wear, and the second being that they were making fun of the leather pampooties that they DID have to wear. Probably was originally a derogatory term for Irish people in general before it came to be applied to the way they speak.
Redwolf
Be suspicious of folk etymologies containing this amount of narrative; they’re usually made up.
As it happens there is a Scots Canadian song as Gàidhlig Chanaideanach from I think the 19th century, and I believe it was collected in Newfoundland by Margaret Bennett, called Òran Nam Mogaisean or “Song of the Moccasins” which is a satire about new Scots immigrants, perhaps composed by the newbies and directed at themselves, and the topic is about trying to make a fist of making moccasins and doing it badly after being shown how by their Micmac (or Mi’kmaq or better yet Lnu if you prefer) neighbors, so the background story goes. The sarcasm in it doesn’t translate directly into English; a straight-up rendering comes out rather flat and affectless.
I’m siding with “mogaisean” being a phonetic approximation of the Algonquian.
Growing up in northeastern Tennessee, we always called our leather farming/work boots “brogans”. Don’t hear that used much these days.
Huh. You know, I remember that, too. Mom and Dad, at least, used the word “brogan” when I was but a sprat, but I seem to recall that the word’s days were already numbered in that neck o’ the woods…er, rather, plains.
I’ve run across the word in books as well…usually to describe a type of leather Oxford.
I’ve always wondered where they got the “-an” suffix. If they took the word directly from Irish, it would be a bit odd, as “brógán” would mean “little shoe” (the “-án” suffix is an older diminutive that functions like the more familiar “-í” suffix). I wonder if, at one point, Irish immigrants used “brógán” to refer to low shoes as opposed to boots, or something like that?
Redwolf
Could be. We always used the term to refer to ankle-high leather boots like these:
Which agrees with [u]this definition[/u].
Interesting. Did you notice they misdefined the Irish word? They were correct in that “brógán” is a diminutive of “bróg,” but “bróg” doesn’t mean “brogue,” it means “shoe”! 
Redwolf
Yeah, brogues were leather Oxfords, and brogans were ankle boots. I recall finding out about the apparent misdefinition a good while back and resignedly figured its adoption must surely have had nothing to do with the Irish themselves.
I think brogans should by definition be flip-flops. Or baby booties. Hey: baby booties look the same as so-called brogans, or at least they used to. I’m not up on that stuff. Anyway, by a long stretch could be it’s a back-usage?
I remember the day when neither bootie nor booty meant “boo-tay”. Bloody irritating how you can’t talk about this stuff any more without risking a raised eyebrow or someone going all juvenile on you and thinking they’re a wit. Sigh.