Different mail servers and different organizations use a variety of different ways to combat spam.
- Various email verification techniques.
- Various mail server verification techniques (ie. reverse lookup for PTR record).
- GeoIP filtering
- Email message content analysis
- Checking the number of "bad addresses" you are trying to send to
- Checking how many emails you have sent to their system
- Checking how many connections you have established with them in the last __ minutes, etc.
The list goes on and on...
The short answer to your question is, "No" you do not need to be sending a constant (or even large) amount of mail to any system to "prove" you're legit.
If you're sending legit email then the best thing you can do is monitor who is blocking you, find out why, and work with them (one on one) to get them to put you on their "accepted senders" list or "whitelist" or whatever they want to call it.
On some systems this is an automated thing... on others it is a human thing. I believe I read somewhere that someone even had an idea of putting up a bond of some sort to prove you weren't a spammer (ie. they catch you spamming you lose the $$$).
If your email list is truly legit there is no reason why you can't use your own mail server and program such as PHPList (or similar products) to send out your newsletters/email, etc.
TALK TO YOUR ISP AHEAD OF TIME and tell them what you're planning on doing.
Even thought your email may be CAN-SPAM compliant, most ISPs have their own unsolicited (or bulk) email policy.
You'll find that most ISPs are very friendly if you talk to them AHEAD OF TIME and tell them your plans. Many will even forward on the "complaints" to you so you can put the complainer on an "opt out" list.