Assuming you're not actually talking about fording a river, but crossing it away from a ford, then the answer is they would have built a pontoon bridge. There are a couple of ways these can be built.

The classic, and sturdiest, of these is to use ordinary boats from the locals and to lay planks along the top to provide the bridge. In some cases, armies carried special boats specifically designed for this purpose, usually when they anticipated needing to build a bridge in a hurry. These purpose built boats could be a simple as specially hollowed out logs, or as complicated as a prefabricated boat that is carried in pieces and then assembled before use. These purpose built boats would have metal hooks designed to rapidly and securely fasten the boats together, rather than a more awkward (and potentially less secure) lashing together using ropes.

Another possibility was to use barrels. Christine de Pizan suggests the existence of planks designed specially for this purpose that could be linked together and place over the top of empty barrels that had been tied together. Philippe de Commynes, although not suggesting any specially type of planking, also makes mention of barrels being used to make a pontoon bridge. In this case, the barrels had been carried in pieces and were put together by coopers as the bridge was constructed.

An earlier variant of the barrel bridge is the skin bridge, made from inflated animal skins. The version proposed to Xenophon by a Rhodian soldier in the Anabasis is the only good description I've found so far - the skins would have been anchored by large stones, then planks would have been laid on top and been covered with earth (to prevent slipping) - but Ammianus Marcellinus mentions bridges made from inflated skins several times, although not how the bladders were covered to allow crossing.

A third variant, mentioned by Christine de Pizan, is simply to use tree trunks covered by planks, much like a series of rafts all connected together. As with the proposed skin bridge in Xenophon, Christine recommends a non-slip layer to be place on top of the planks (in this case reeds and manure rather than earth).

Bibliography

• De Re Militari, by Vegetius, tr. N. P. Milner
• The Book of Deeds of Arms and of Chivalry, by Christine de Pizan, tr. Summer Willard
• The Roman History of Ammianus Marcellinus, tr. J. C. Rolfe
• Maurice's Strategikon: Handbook of Byzantine Military Strategy tr. George T. Dennis
• • Three Byzantine Military Treatises*, by George T. Dennis
• Philippe de Commynes, Memoirs, tr. Michael Jones
• Anabasis, by Xenophon, tr. H. G. Dakyns

If you haven't already found them, then I would also recommend you check out David Nicolle's Medieval Warfare Source Book, Volume I & II, Clifford J. Rogers' Soldier's Lives Through History: The Middle Ages and Warfare and Society in the Barbarian West, by Guy Halsall. The last is based on your mention of both the middle ages and the migration period and my assumption that you're setting your story in the early middle ages, but if you're planning to set it earlier or later than that, let me know and I'll suggest something more appropriate to your planned era.

Edit: also, while I think of it, J. Kim Siddorn's Viking Weapons & Warfare has a lot of practical information about fighting in a shield wall that's relevant to earlier and later periods as well.