I bike to work in Japan, where we still have lame archaic office dress code requirements (the best I can do is Brooks Brothers chino pants and a tucked-in open-collar dress shirt with a belt).
It's an uphill ride to work, so I bought a mountain bike with one of those electric assist motors to help with the sweat thing. It makes riding up a moderate hill no more taxing than riding a regular bike on flat ground.
I thought it would be kinda lame for regular bike purposes, but it actually works out well. The law here says the motor has to cut out at 25kph (16mph). And when that happens, because it is really heavy with the huge battery and motor, it becomes extremely taxing to coax any more speed out of it. But riding around at max speed feels still gets my breathing and pulse up.
So the upshot is that I still get a lot of moderate exercise from the bike, and I can locomote around Tokyo at a pretty much constant 25kph, using either the road or the sidewalk depending on traffic/pedestrian conditions.
And, because I can take a pretty direct route to work, I get there in half the time it takes on the subway.
IME most workplaces have showers. It pays to ask; they can be well-hidden. Failing that, ride at an easy pace in the morning, wear breathable clothes etc. Cycling needn't be any harder than walking (unless you have huge hills or something).
Many companies here (though I guess I only really know larger corporations) ensure there's adequate shower facilities for washing down and changing in to business attire before work. In my new office's case, the office catering supplies fresh towels that you just throw in a bin after using and they look after it all.
We can also leave our details with someone here and they'll arrange for a bike service to be performed during the work day.
I've noticed a recent up-tick in companies (especially engineering firms - one of them sponsors the major cycling event in this country) embracing cycling as an activity and mode of transport. It's really good to see.
The building I'm working in (there are multiple companies and the building is managed) has showers, so it is quite convenient. The only annoying bit is that there are no lockers, so I pre-dry my sports gear with a heardryer (this is actually quite quick, takes two minutes to nearly completely dry three layers), and then keep it under my desk. Might look bit "unprofessional", but once everyone accepts the slightly "eccentric" behaviour ("did you really cycle in today? In the snow?! crazy.."), it's not that bad.
