Hi, Mijail (did I say that right?) Your question is specific to business, and so I'm answering in that capacity. Yes, I think you should choose an "American Business" name. In the marketplace, having to explain, spell, and re-pronounce your name is as counterproductive as having to do that for your product or corporate name. "Misha" is okay, but it's a half-measure. Unless you're building a business that is somehow built off of your own personality and identity, then you probably want your own personal name to fade into transparency in your business interactions.
Your last name is probably different or interesting enough to keep you from fading into the ocean of Mike Smiths and Bob Johnsons in the US. I was always proud around the office that I could spell and rattle off the name of my old associate Andrzej Olszewski, but the reality was, in business, he went by "Anj", and took up space on his business card to articulate the pronunciation (an-jay). The fact that he is a naming and branding expert almost made it a positive conversation point for him, but he was usually working within corporate structures where his "exotic" background as a scholar from Poland was a big plus. But if you're an independent business person or founder or salesman, or have any other kind of role where you're introducing yourself and needing other people to be able to talk about you and to you and remember your name, and your personal identity is a critical part of your brand, I think you should go all the way with a name that works just like the name for a product or company - simple, easy to spell, clearly understood on the phone, handy to use in conversation. And so, if you're going to use something besides your name, go all the way to the client and make it useful and handy for *them*, just as you would a product name.
Rather than Misha, I'd consider some other, related names that make sense to American/English ears - especially a name that is also a word in the language: Mike, Max, Matt, Mark (!), Mac, or Miles. Another option might be a common American nickname with a little snap, which few people will assume is your given name - Red, Stretch, Buddy, Junior, Pops, Shorty - or, something derived from your last name that's easy to say, that people will assume is NOT your real first name - US examples would be like Smitty, Jonesy, Bake. If your last name was Razodan, you could be "Raz" Razodan. If your last name was Combunkos, you could be "Com" or "Bunk" Combunkos. But I think your best bet would be in the Mike, Max, Mark category.
As far as the name on the check, you have a few options: 1. Change or add to your name at the bank, it's probably a simple enough matter to get a name added as a parenthetical to your identity there, such as Mijail "Mike" Razodan (or whatever), then a check made either way will usually work, and you could get the name added to your next batch of checks. 2. Always specify on your invoices how the check should be made out - I run my business as an individual, Mark Gunnion, but in some of my promotion, I refer to my business as Mark Gunnion Names. But on my invoices, I always say specifically, at the end, "Make checks payable to Mark Gunnion". 3. If you use Mike or another Anglicized name starting with M for business, you could avoid muddying the communication by saying "Make checks payable to M. Razodan."
If you go with the Anglicized name version, Max, Mark, Mike, choose one where the last sound of the first name is different from the first sound of the last name. Max Tierra, but not Max Saud. Mike Tyson, but not Matt Tyson. Mark Baker, but not Mark Copeland. This will make the name work better in audio situations, in conversation or on the phone. Even my name, with a "k" going into a "g", can be confusing. When I was in the Boy Scouts, for example, after the first time they heard my name said out loud - Mark Gunnion - for years after, they all called me Mark Onion!