I have a bunch of motors and sensors from the original Mindstorms set. It seems a shame I can't use these with the more recent NXT set. Are there any unofficial modifications that allow you to convert the cables?
LEGO used to sell Converter Cables for Mindstorms NXT at a price of £8.99 / US$9.99 for 3 cables at their online shop, they can still be bought online from stores like Bricklink.
Considering the price of each cable converter it may be worth purchasing newer sensors/motors that are more advanced. The NXT system cables are separate from sensors and motors and come in different lengths.
Yes, you can use all RCX sensors and motors with a NXT brick. Don't forget about lamps, too.
The connections are different though, so as Ambo100 says, you will need converter cables. However, if you don't want to pay the full price for them, you can also make some yourself which isn't really that difficult.
The other thing you need to consider is how to use them with your programming environment; for the LEGO-provided NXT-G there is a special library of blocks which you can add. You can find it in the patches section fo the Mindstorms support page, look for "Legacy block". The description of these blocks is as follows:
On a related note, if you want to use Power Functions motors and lights, you can do so by using the two converter cables (NXT-9V + 9V-PF). There is currently no NXT-PF conversion cable, but you could make one as easily as you would make a NXT-9V cable. There is also a third-party infrared link "sensor" which allows an NXT brick to control PF elements via the normal infrared receiver.