Are there any countries that allow gay marriage that don't have a longish history of Christianity?

No. There are 17 countries that allow it and 2 that allow it in some jurisdictions. A list may be found here: http://www.pewforum.org/2013/12/19/gay-marriage-around-the-world-2013/

There have been plenty of cultures where homosexuality was accepted; classical Greece and Rome, for example. Cultures where marriage is predominantly a governmental matter rather than a religious one are all, as far as I am aware, heavily influenced by the cultures of western Europe. One might also observe that all of these countries are industrial or post-industrial, and have large populations of young people with vastly more economic and sexual freedom than occured before the middle of the 20th century. One might also observe that China, Japan and South Korea seem to be the only countries at this level of economic development that were not culturally dominated by colonial states.

The fact that a history of Christianity is positively correlated with approval for gay marriage does not imply that Christian memes directly influence stances on homosexuality. Christianity spread around the world alongside other memes (such as democracy and case law). Those countries where European colonies were culturally dominant also received the industrial revolution and the immense increases in personal rights that came as a consequence of the increased economic and political power of the working class. One might also point out that thinking black people are inferior is a meme that arose from the slave trade in Christian semi-democracies.

There seems to be abundant evidence that the Abrahamic religions have strongly influenced societal views worldwide with regard to sexual morals; indeed, I cannot imagine a remotely plausible argument for this being untrue. I also wish to observe that Eastern Orthodox Christianity survived the USSR and still affects cultural values in Russia; it seems highly improbable that it did not influence Russian culture in the 1930s.