When people talk about alliterative verse, they usually mean Beowulf. It is the most familiar piece of alliterative poetry, at least for speakers of English, who often encounter it in school (at least for a week or so during a quick survey of British literature.) So that is what we will start with.
A 19th Century scholar, Eduard Sievers, came up with the classical description of the rhythm of Beowulf and other Anglo-Saxon poems. He observed that the half-lines of Old English poetry come in five basic rhythms.
The following proverb-like little poem illustrates the five types:
Pride and anger brought pain and loss,and hate festered. Hell's masterpieceoverwhelmed all.
Type A: Lift-Dip, Lift-Dip
For example: pride and anger
Type B: Dip-Lift, Dip-Lift
For example: brought pain and loss,
Type C: Dip-Lift, Lift-Dip
For example: and hate festered.
Type D: Lift, Lift-Dip (with a secondary stress in the dip)
For example: Hell's masterpiece
Type E: Lift-Dip-Lift (with a secondary stress in the dip)
For example: overwhelmed all.
These are the basic rhythms; there are variations on them, but nearly every line of Beowulf fits into these five types and a small set of normal variations.