I've perused a couple of these kinds of sites. The major sites are cafepress.com, printfection.com, zazzle.com and spreadshirt.com.
The first three are all based in the US while spreadshirt is based in germany (meaning lower cost of delivery for us europeans). I've only tried spreadshirt, because I don't have a lot of money to throw around sadly, and it was satisfactory, though I used a gradient in my image and that didn't show up too well if you look close. The price was decent too.
From what I've heard zazzle is supposed to have good quality on their stuff, but they are also said to be the most expensive of the bunch (their swedish site is horrible too. they used google translate or something and I guess they forgot that crowns is not 1:1 to dollars)
I'd say if you're in the US try zazzle, if you're in Europe, try spreadshirt.