Despite the present hostilities, at a cultural level, some Russians have made an effort to bring Georgian culture to Russia.
For instance, the major Russian poet Arseny Tarkovksy, father of the famous film-maker, translated poetry by several Georgian authors into Russian. In a book of his own poems and of his translations, you can find poetry by Grigol Orbeliani, Rafael Eristavi, Ilya Chavchavadze, Vazha Pshavela, Galaktion Tabidze, Georgi Leonidze, Simon Chikovani, Karlo Kaladze, Irakli Abashidze, Revaz Margiani, Yosif Noneshvili and Mikhael Kvlividze.
No, I'd neither heard of these poets before (bar one), and haven't read Tarkovsky's translations as my Russian isn't really good enough. But it shows that there is a whole range of 19th and 20th century Georgian poets whose work we in the West know nothing about.
This German article gives an overview of Georgian literature as a whole:
Georgische Literatur
And a book of 20th century stories:
Amazon.de: Georgische Erzählungen des 20. Jahrhunderts: Naira Gelaschwili: Bücher
One author:
http://en.wikipedia.org/wiki/Ilia_Chavchavadze
(Oh, and please don't bring up the sickly fact that Simon Sebag Montefiore has written about Stalin
the poet. Minor poet, major mass-murderer. Hitler painted the odd picture. Both did their bit for culture.)