Kukushkin is ranked 58th in the world rankings and has lost to Murray on both the two occasions they have previously played each other.
Murray, the 2013 champion, could meet 10th seed Nadal in the last eight but was drawn in the opposite side of the draw to Novak Djokovic, meaning the Scot cannot meet the world number one until the final.
Avoiding Djokovic will come as a relief to Murray but his route to the final is far from easy with Jo-Wilfried Tsonga potentially waiting in the round of 16 and he could face seven-time champion Roger Federer in the last four.
Murray has already reached the Australian Open final and French Open semi-finals this year, both times losing to Djokovic.
Last year's Wimbledon in pictures:
The British number one followed up his best ever clay-court season by winning the title at Queen's on Sunday and he will be confident of bettering his performance last year at SW19, when he lost to Grigor Dimitrov in the quarter-finals.
Djokovic remains the principal threat to Murray claiming a second Wimbledon crown but the top seed faces a tricky opening encounter against Philipp Kohlschreiber and may have to beat Lleyton Hewitt, Bernard Tomic and Kevin Anderson - all dangerous opponents on grass - before the quarter-finals.
The world number one is due to meet fourth seed Stan Wawrinka in the last four, the same opponent he lost to at the French Open final only three weeks ago.
Federer, who is bidding for a record eighth Wimbledon triumph, will face Damir Dzumhur in the first round while Nadal plays Thomaz Bellucci and Wawrinka is up against Portugal's Joao Sousa.