Amazon does not have an international branch in India (it has in China, France, Canada, etc.etc.)
However, www.amazon.com does ship new books to India.
In the case of used books, that are sold by independent sellers as Amazon Merchants,
you must check the single seller. Some seller ship internationally, others do not do. In any case I think that the best way to pay is by Credit Card. I do not think anybody accepts "cash" anymore and if they do the shipment of cash is quite a bet in the dark. Some sellers accept Paypal. Cashing a foreigner check is usually an expensive and risky operation so it is difficult that a seller accepts checks.
(I'm speaking from experience, since I'm Italian and I bought about 500 books from USA in the past few years.)
You can also look for your books on
www.bookfinder.com
there can be the chance that their are available from some seller in India, or that you can find other sellers willing to ship to India.