Since the Lions returned from the Republic 12 months ago, Wales have faced the giants of the southern hemisphere on five occasions and are yet to record a win.
Wales were beaten by both the All Blacks and Wallabies at the Millennium Stadium in November and then by the Springboks at the same venue in early June.
A trip to New Zealand and two Test defeats against the number one ranked team in the world followed shortly after, leaving Wales with just one win over any of the big three since 2006.
But McGeechan, who has led Lions tours to all three super powers, is confident that Wales can mount a serious challenge when Australia, New Zealand and South Africa arrive in Cardiff in November, if they stay positive and believe in themselves.
"I think that sometimes the Welsh psyche is to do well in certain games and then hold off a bit," said McGeechan.
"So it is a case of getting hold of the Welsh psyche and saying, 'Look, we are up there. We should be looking at beating the southern hemisphere teams.'
"Wales know their game has to be top drawer. They are trying to play an attractive brand of rugby and have the players to run in tries from anywhere.
"But controlling the field position and possession is that little bit more important and Wales have to build up time with the ball.
"Wales were competitive in that last Test against New Zealand and I think they will have taken something from that game.
"They will think they can move forwards with what they have done in their last Test. The bit of luck that you need is your best players fit."
McGeechan also chose five members of the Welsh coaching staff to assist him with the Lions in South Africa, with Warren Gatland, Shaun Edwards, Rob Howley, Craig White and Rhys Long all touring with Britain and Ireland's elite.
And having built a special relationship with those coaches, McGeechan has no doubt that Gatland and co are the right men to take Wales onwards and upwards.
"Wales have a great coaching team and I really enjoyed working with them when they were with me with the Lions," added McGeechan.
"Warren, Shaun Edwards and Rob Howley have given the Welsh players a very clear perspective in everything they do.
"Wales have got good players right through the team and I think 'Gats' is astute enough to challenge them."