How did Old English poets build a line without rhyme?

With repeated opening sounds instead of repeated endings. Old English poetry belongs to alliterative verse, a form that uses alliteration as the principal device to mark its metrical structure, rather than rhyme. Each line is split by a pause, a caesura, into two half-lines, and the two halves are linked by alliteration: the first stressed syllable of each half must begin with the same sound as the first stressed syllable of the other. Each half-line usually has two heavily stressed syllables, called lifts, and the lighter syllables between them are dips.

The rules are strict. Only stressed syllables of stressed words, most consistently nouns and adjectives, can alliterate. Consonants alliterate with the same consonant, with sc, st and sp alliterating only with themselves, but a vowel alliterates with any other vowel. And this is not syllable counting: a dip may be one unstressed syllable or a whole string of them. The famous first verse of Beowulf, Hwæt wē Gār-Dena, "we indeed have heard about the Spear-Danes' (power)", is an example of a verse.

The constraint shaped the poets' language. The need to find a word with the right opening sound gave the tradition a specialised vocabulary of poetic synonyms and standard metaphors called kennings, and Anglo-Saxon poets used epithets frequently to achieve the desired alliteration. All of it was composed and passed on by voice, and much went unrecorded.

After the Norman conquest, French forms with their syllabic emphasis gained prominence, and this kind of verse lost its dominant place in English. It did not die: later alliterative poems such as Piers Plowman and Sir Gawain and the Green Knight use it, and modern authors, including W. H. Auden, have revived it.