Could you try H & M? I dispaired of this for dd but found several nice, inexpensive, completelyplain t shirts for her there.
If online is essential, then Mini-Boden do them. Their stuff is expensive but does last and last and last.
Jako-o have loads of logo-free stuff. Slightly complicated by their being abroad but I don't think it's a problem.
(LL Bean are nice, but their shipping is expensive and I think you are more likely to have to pay duty etc. I nearly bought something from their catalogue once but when I looked at the shipping details it put me off.)
Thanks - I have been to Jako (from previous MN reccommendation!) but they didn't have one of the sizes I wanted - I am aiming for the cute "both DSs in the same outfit for weddings etc" look. I have their shorts coming fromk Next and nice checked shirts but wanted T shirts for underneath. I will check out this US one. Or, probably end up in tesco!!!