Mowing like a (backyard) beekeeper requires, first, that one divest themselves of any remnants of a "keeping up with the Jones" mentality. Your yard is that...a yard, a pasture, not a "lawn" to be manicured.
Which leads to the second point. You have to utterly disregard how the neighbors are going to perceive you. "Dandelions...just going amok! Haven't they heard of Round-up??" (By the way, you will be greeted with a shotgun if you try to spray my beloved yellow weed-flowers. Spray 'em in your own yard. Over here, I'm propagating them. Seriously. Bees love them...and dandelion honey has such a delicate, sweet flavor!) There is also a bit of a jungle around the front of the house. Well, I'm just waiting for the Star of David to bloom. Once they are done, I'll mow it.
Third, you will learn to dodge bees and clumps of flowers. This will result in wiggly, scriggly mow lines. If you want a checkerboard patterned lawn--don't become a beekeeper.
Similarly, you will just laugh when you see a low-flying plane and question whether the pilot thinks the idiot downstairs is mowing drunk.
Lastly, alternate which parts of the yard you do mow...because you really don't want the grass to eat your property.
How to mow like a beekeeper: Mow the grass...miss as many flowers as possible...and ignore what the neighbors might think.
Your crazy resident beekeeper,