McGeechan's predecessor, Warren Gatland, masterminded a Premiership title hat-trick and Heineken Cup final triumph, but Wasps missed out in both competitions last term.
Hopes of a record-equalling fourth successive league crown were dashed by semi-final opponents Sale Sharks, while Wasps failed to emerge from a tough Heineken Cup qualifying group dominated by French giants Toulouse.
An Anglo-Welsh Powergen Cup final triumph on a dank day at Twickenham was scant consolation for a poor campaign by Wasps' standards, and there is a burning desire to re-establish normal service.
"Playing at Twickenham in front of a good crowd will be a big motivator for all the players," said McGeechan, ahead of a London 'double header' clash against Saracens.
"The Premiership is getting more and more competitive with every year that goes by, and therefore it becomes more important to get off to a good start.
"We know we have to hit the ground running, but facing a good side like Saracens it's going to be tough right from the off."
Former Munster chief Alan Gaffney takes charge of Saracens for the first time in Premiership combat, fielding a side which features newcomers Tomas de Vedia, Fabio Ongaro and Paul Gustard, but former Great Britain rugby league captain Andy Farrell's long-awaited union debut remains on hold.
Gaffney is well aware of Saracens' status as Premiership under-achievers during recent campaigns, and they will begin another season as little more than a mid-table chance in most pundits' eyes.
"The Wasps game will give us a good idea of where we are," he said.
"We are confident we will put on a good display, but what we must do is concentrate on our own performance and the work we have been doing pre-season."
Harlequins boss Dean Richards has targeted an "immediate impact" from his team on their Premiership return after last season's title-winning sojourn in National League One.
Quins have recruited heavily - Hal Luscombe, Stuart Abbott, Nicolas Spanghero and skipper Paul Volley all make debuts in the Twickenham opener against London Irish - and know there can be little time to familiarise themselves with top-flight rugby again.
"We are fully aware of the step up in performance that will be required in order to compete at this level," said Richards.
"We have worked incredibly hard during the summer to make sure we are fully prepared to make an immediate impact.
"Having achieved our objective of returning to the Premiership at the first attempt, we are now looking forward to taking on the best clubs in the country once again."
Irish, last season's surprise package, field 12 of the side that started against Premiership title play-off opponents Leicester four months ago, the only exceptions being prop Tonga Lea'eatoa, hooker Danie Coetzee and lock Nick Kennedy.
Bristol avoided relegation last term thanks to a succession of wonderfully-resilient displays, and head coach Richard Hill will expect to see those qualities on show at Worcester.
Centres Brian Lima and Sam Cox are both sidelined through injury, but scrum-half Shaun Perry is back after missing England's summer tour to Australia and 37-year-old Welsh veteran Gareth Llewellyn - currently the Premiership's oldest player - also starts.
"We are not a team which relies on one or two special players to provide a spark," said Hill.
"We work as a team, and all our 40-odd squad members know our system pretty well and can slot in. Missing one cog is not vital, because someone else can slot in and do a good job."
Champions Sale, firm favourites to emulate Leicester and Wasps in retaining a Premiership title during rugby's professional era, kick off against Tigers at Welford Road on Sunday.
Sale smashed Leicester 45-20 in last season's Premiership final at Twickenham to take silverware back to Stockport, meaning Tigers have now gone four years without landing a domestic trophy.
Jonny Wilkinson, meanwhile, sets out on the road towards next year's World Cup by starting for Newcastle at Northampton on Sunday, where he will encounter Saints' genial fly-half Carlos Spencer.