if you dont have a pair on the flop you have 6 outs to pair on the turn, there are 47 cards left in the deck (52 - 2 (your hand) - 3 (the cards on the flop). So your chance of hitting a pair on the turn are 6/47 or 12,8%.
for a roughly estamation you can multiply the number of outs by 2 to get the % for hitting on the next street.

It might be because I just woke up, but I was thinking this would be 6/50 + 6/49 + 6/48 + 6/47 + 6/46 but that comes out to like 60% and would come out to over 100% in a game with more cards so I'm clearly doing something wrong.
I'll probably figure it out later because its bothering me, but you might be able to figure it out using this:
http://stattrek.com/Tables/Hypergeometric.aspx
Just keep in mind that if you put like 50/5/6/1 I'm pretty sure that it is calculating the chance of hitting EXACTLY one of your cards and not including pairs on the table.
Edit: I figured it out, new post coming.

6/50 = .12
6/49 = .122
6/48 = .125
6/47 = .128
6/46 = .13
Now using the odds that you DON'T hit these..
.88 x .878 x .875 = 67.6% you will miss the flop = 32.4% chance you will have hit
x .872 = 60% chance you will miss by the turn, 40% chance you will have hit
x .87 = 51.3% chance you will miss by the river,  48.7% chance you will have hit
This doesn't include the board pairing.

FlopZilla is great for getting an idea of this kind of thing. It's free, try it. You can enter more cards on the flop section to understand how different boards hit different ranges all the way to the river.
http://stoxev.com/flopzilla/flopzilla.htm

This is understated if you want to include the possibility of 2 pair on non-paired boards (which I can't imagine that you would not want to). With 2p included turn is 46% (40.7% exactly one pair) and river is 56% (47% exactly one pair).
The most effective way to see this from the perspective of playing cards is to download flopzilla, select all cards except pocket pairs for range. Select board cards that do not allow for straight or flush (eg, 2h 6c 9c Jd Ah) - first select 3, for flop - then add one, then add one. I tried to post a couple screen shots but don't know how.
Add the pocket pairs back in, pick more reasonable ranges, try different boards ... Seriously, if you want to learn about board texture there is no better way.