Absolutely! In fact you should design your logo with this in mind. There's so many poorly marks that don't hold up these days.

It's not just about how it looks on the internet! People still do use Xerox machines these days believe it or not. At that point it's black and white. I know, ancient technology! =)

Seriously though. Think about the following cases:
- Business cards (small print)
- Footers on letterhead
- Billboards (massive size)
- Photocopies (and copies of copies)
- Home printers (bleeding ink, low resolution, bad paper)
- Like you said, t-shirts and other cloth or vinyl or plastic or ceramic...

If your mark holds up to all that...You need not worry so much about how it looks on screen.

Just don't go wild with the color. I mean some brands do, right? You see logos in all sorts of color. It's the mark that matters. However, I'd suggest having an overall color scheme you stick to. You should have a brand style guide and it should be used and given to all designers for all purposes from web to print.