[Preflop]

There are a couple issues you need to think about when betting the river, and most of them come down to how well you know your opponent. Assuming he is at least an average player, he will be able to throw that set away to almost any river bet.

One situation is that he flopped the low end of the straight with a T 8, and decided to call you down. In this case his straight is actually higher than yours, and he would surely not fold to a river bet after calling the flop and turn, and would win the pot.

You can't discount the possibility that he flopped the nuts with KT also. The reason I see this as a possibility is because of the nature of the flop. There are a lot of cards that change the nuts, and with no redraws, it is much safer to simply call your flop bet with position, and depending on how tricky and/or cautious your opponent is, flat call the turn as well. In this situation a river bet definitly hurts you.

The only real downside from checking the river is that your opponent may take this as a sign that this pot is up for grabs. If your opponent is tricky enough, not betting the river could cost you the pot, because imho a pot sized bet is very hard to call on that river with what your holding.

Hope that helped.