Ready for my brilliant response?
What I do is simply forward all email that comes to me @me.com account to my Gmail account. Problem solved, plus you're not checking email in two places.
Just login to iCloud.com, then go to Mail, then click the little gear in the top right corner, Preferences... and on the first tab there's an option that says "Forward all mail to..." and simply put your Gmail account in there. Below that is the option to delete any mail after it is forwarded or not (checkbox). That's up to you. If you want to set up particular rules in Gmail, just make them if SENDER is "account@me.com", then either send it to this folder or do this or that.
As someone else pointed out to me on the (official) Apple forums, @me.com addresses also accept the +tag convention that Gmail does, which is awesome. So you can tell someone to send you an email at "account+test@me.com" and it will go to "account@me.com"! When it forwards, I believe it will also come from "account+test@me.com" for which you can setup specific rules -- either in iCloud itself (which also has its own rules section that you could use to possibly "whitelist" senders") or just use Gmail itself.
Personally, if something is a newsletter, I'll put in my email as like "account+newsletter@me.com" (or @gmail.com) and I can sort it to a specific folder, etc from that. For instance, the first rule I would set up after forwarding to Gmail would be "any mail from *@me.com" --- "do not mark as spam", "apply label (whatever, if you want it to label/tag those)" etc.
You can really use the +tag to your advantage. If you KNOW something is gonna be junk that you don't want, I have a special "account+spam@gmail.com" rule that says "mark as read, send to spam folder (or directly to the trash), bypass the inbox". Likewise, for important emails I *never* want to miss, I use the tag "account+notspam@gmail.com" and add the rule "never send this to spam" to ensure that never happens.
Be creative with it, I have a bunch of special uses, such as:
+alert (star it, mark it as important, never send to spam, maybe forward it to a special account that only emergency emails are supposed to go to)
+newsletter (usually bypass inbox, go to Newsletters/whatever)
+urgent (will be starred and marked as important, never sent to spam, pretty much identical to +alert from above)
+receipt (for all my iTunes purchases, especially the free ones, it gets annoying to get an email after every app I download, so I have them set to automatically bypass my Inbox and go directly to Receipts/iTunes label "folder" if the sender is "whatever@whatever.itunes.com")
The possibilities are almost endless with this system. Also, with Gmail, if someone asks for your email, but you don't REALLY want them to have it long term, I believe you can always use a period (hard stop) in the email address anywhere, and it will still be delivered to you, BUT you can always set up a rule in the future to automatically delete any emails sent to that address! For me, I usually give my friends "firstnamelastname@gmail.com", and those who I just want to use temporarily I'll give them "firstname.lastname@gmail.com" which appears to most people to be a perfectly legit email address and they won't question it, plus you'll still receive it (if you want).
Give all that a shot, let me know if that's a good solution for you. I hope I could help out! I know that's a frustrating problem to have.
(I want to note, although MOST email systems allow the + in your email address, there ARE some sites that refuse it as a valid email. Just be aware of that possibility. For THOSE instances, that might be a case where you could alternatively add a period in your email, and use that as a filter, as this is pretty much universally accepted. So you know if you have email coming to "firstname.lastname@gmail.com" to process it according to whatever logic makes sense to you. Good luck!)
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