Yes. This seems to be the consensus (from Wikipedia):
"Many countries claim to have invented the chess game in some incipient form. The most commonly held belief is that chess originated in India, where it was called Chaturanga. The earliest mention of Chaturanga appears in the Indian classic, the Mahabharata, written circa 2,000 BC. Another theory exists that chess arose from the similar game of Chinese chess, or at least a predecessor thereof, existing in China since the 2nd century BC. Joseph Needham and David Li are two of many scholars who have favored this theory.
Chess eventually spread westward to Europe and eastward as far as Japan, spawning variants as it went. One theory suggests that it migrated from India to Persia, where its terminology was translated into Persian, and its name changed to chatrang. The entrance of chess into Europe, notably, is marked by a massive improvement in the powers of the queen. The oldest known texts describing chess seem to indicate a bi-directional spread from the Persian empire. From Persia it entered the Islamic world, where the names of its pieces largely remained in their Persian forms in early Islamic times. Its name became shatranj, which continued in Spanish as ajedrez and in Greek as zatrikion, but in most of Europe was replaced by versions of the Persian word shâh = "king".
spawning variants is the key word...modern chess as we know it came from Germany