If spammy emails are getting posted to your Tumblr blog against your will, here’s a possible fix*:
- From your Tumblr Dashboard click on the name of your effected blog.
- Click “Blog Settings”.
- Scroll down to “Post by Email”.
- Click the “Reset” button.
- Click the “Save” button.
- If you use the Post by Email function, make sure to update your address book.
* I do not know with 100% certainty that the Post by Email feature is the culprit. Your spammy problem may be due to a different cause. It’s worth giving a shot though. Good luck.