Kukushkin is ranked 58th in the world rankings and has lost to Murray on both the occasions they have previously played each other.
“He’s a good player, he’s had some big tournaments in the past,” Murray said.
“I think he played Rafa (Nadal) pretty close here last year, I think it was five sets, and he can play well on the grass.
“He’s a good player. He’s plays predominantly from the baseline, but he’s a good ball striker and if you give him time to dictate the points he’s a tough guy to beat.”
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 No 1 before 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.
The British No 1 followed up his best 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 No 1 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.
There were difficult draws for Britain’s other male contenders as James Ward was pitted against Spaniard David Ferrer, whom he used to hit with during four years training in Spain.
“It’s obviously a tough one,” Ward said. “He’s a good player and it’s going to be a very difficult match. When I lived in Spain we played a little bit because we lived not far away from each other so I know him pretty well and I know what to expect.
“Can I hurt him on grass? Of course. He’s played well here in the past but if I go out there and play my game, who knows what can happen?”
Yorkshire 20-year-old Kyle Edmund, who impressed at the French Open last month, is up against Ukraine’s Alexandr Dolgopolov while Aljaz Bedene will face experienced Czech Radek Stepanek.
Liam Broady plays Marinko Matosevic and Brydan Klein will meet Italian 25th seed Andreas Seppi.
In the women’s draw, British No 2 Johanna Konta was handed a fearsome opening match against fourth seed and 2004 champion Maria Sharapova.
Konta arrives at the All England Club on the back of a superb run at Eastbourne this week, when she reached the quarter-finals thanks to victories over two top-20 players.
“I’m looking forward to it and excited to play Maria since I’ve never played her before,” Konta wrote on Twitter.
“Because of Sharapova’s profile, the match hopefully should be on a big court.
“This is what every tennis player dreams of, competing on a big stage, against the best and it’s even better that it’s at home at Wimbledon.
“I believe in my ability, I’ve absolutely nothing to lose and I’m going to go out there and do everything I can.”
Sharapova is in the same half as top seed Serena Williams, who is gunning for her sixth title at Wimbledon and her 21st grand slam overall. The American plays Russia’s Margarita Gasparyan in the first round.
British No 1 Heather Watson could meet Williams in the third round but she will first have to come through a first-round clash against France’s Caroline Garcia.
Laura Robson, who has been given a wild card after 17 months out with a wrist injury, faces Russian Evgeniya Rodina and British No 3 Naomi Broady plays Colombian Mariana Duque-Marino.
America’s Sam Querrey will play Uzbekistan’s Denis Istomin in the final of the Aegon Open as both players continued their Wimbledon preparations with semi-final victories.
Querrey, who is ranked 44th in the world, came from a set down to beat Ukraine’s Alexandr Dolgopolov 4-6 6-3 7-5 in Nottingham while Istomin also progressed to today’s final after defending champion Marcos Baghdatis retired with a calf injury.
Querrey is looking to win his first ATP title since 2012 and he will take heart from his performance against Dolgopolov, who overcame Rafael Nadal at Queen’s last week.
Baghdatis had beaten David Ferrer earlier in the tournament and will decide this weekend whether he is fit enough to play in SW19 after he was forced to pull out with a 2-1 lead over Istomin in the first set.
The 2006 Wimbledon semi-finalist is scheduled to face America’s Donald Young in the first round at Wimbledon.
Istomin, ranked 92nd in the world, has lost four out of his last five meetings against Querrey.