Why don’t we write poetry like Beowulf any longer?

Solution 1:

Literacy, pens, paper, the printing press.

A written culture has different restrictions than an oral culture dependant on ease of repetition from memory.

According to the University of Wisconsin Digital Collections Center:

Beowulf is the oldest narrative poem in the English language, embodying historical traditions that go back to actual events and personages in fifth- and sixth-century Scandinavia. During the long preliterate centuries when these traditions were transmitted in the form of oral poetry, they were combined with with a number of legendary and folktale elements (among these are Grendel and his mother, the dragon, and probably the hero Beowulf himself). The written text of the poem, as we have it today, took shape in England during the middle or late Anglo-Saxon period and survives in a single manuscript from around the year 1000.

An oral tradition requires stories to be easily memorised and stand repetition many, many times, and passed on to the next storyteller. A strong metre and fixed structure with helps, along with alliteration (also found in Beowulf) and isn't unique to English. For example, the Finnish national epic, the Kalevala, is based on sung oral tradition and has a fixed pattern of stresses, with much alliteration and parallelism (repeating the previous line with different words but meaning the same thing). In fact, the hero, "steady old Väinämöinen" (an almost always alliterative "vaka vanha Väinämöinen"), is himself a storytelling wizard who plays a zither and uses his song-words for magic.

Once people can read and write, they no longer have a need for the storyteller to recite a story from memory, they can read it themselves, or have someone read it to them from a text. Over time, this gives rise to more creative ways of expression.

For more, here's a paper (PDF) on Oral Tradition & Its Decline by Indira Bagchi.

Solution 2:

Who says we don't? Have you listened to rap or hip-hop lately? Anglo-Saxon poetry like Beowulf was heavily beat-based and while it didn't involve rhyme it used alliteration that gave similar aural cues. The lines were recited four stressed beats to a line with a caesura dividing it into two-beat groups, and rhythm was important. I have long considered Anglo-Saxon poetry to be the rap music of its day.