What motivated Germanic tribes to accept Christianity?

What motivated the Kings of pagan tribes to move to Christianity? Christianity was more compatible with Hierarchy and structures the relatively unlimited power of divine right kings rather than the relativity more democratic or meritocratic pagan traditions. The medieval church was a good ally of Kings. A certain congruence of values between Kings and Bishops.

The adoption of Christianity by Germanic tribes was an event driven by the Political leadership. Thus the answer lies in what advantages did Christianity have over paganism for this leadership? This at a time when the leadership of these tribes is pushing towards a more centralized and powerful Kingship, Christianity (as the early medieval church) was more compatible and more philosophically inclined to support the more centralized King's power than paganism.

One important thing to keep in mind is that Religion is a marker of culture.

The Goths, Franks, and Vandals converted for rather practical reasons. They had conquered territories from Rome (modern Italy, Spain, France, and North Africa) where the populations they were trying to rule were all Christian. They were never more than a ruling elite in these areas, and from a financial standpoint had everything to gain and little to lose by pacifying their subjects as quickly as possible. Attempting to wipe out the local culture when they took over would have guaranteed strife for generations, and might have destroyed much of the machinery that kept the goodies flowing upward.

So what they did instead is embrace the currently fashionable heresy (Arianism), allowing them to install their own local pope answerable to them. The exception there is the Franks, who managed to make themselves powerful enough in France that they had nothing to fear from owing liturgical allegiance to a foreign pope.

The second important factor I think is the cultural bloc that Christianity represented. At this point you have the central and most populous and prosperous kingdoms in Europe being Christian. Fellow Christian kingdoms would have enjoyed preferences in trade and international relations that were quite compelling. It would have been much like being part of the EU today. For peripheral areas like England, Scandinavia, and Eastern Europe, the trade benefits alone would have made it worth it. This is why Eastern Europe areas that tended to have stronger trading ties with Constantinople generally converted to Eastern Orthodoxy rather than Western Europe's Catholicism.

I would imagine not being a perpetual target for Crusades also had its attractions.

As far as I know, for a Germanic tribe to immediatly convert to Christianity did not explicitly mean that all of the tribe members would convert, in fact, most of the time only their leaders would, and it would be enough for the parties which demanded the conversion to be satisfied: here I am clearly implying that converting to Christianity would most likely be not a self-taken decision (unless you were as crazy as e.g. Constantine) but mainly done for political or diplomatic reasons. Moreover, it would probably be only symbolic, and traditional pagan customs would still be performed and even merged into the practice of the Christian denomination of choice.

Let's imagine and place ourselves around the middle of the fourth century. Rome is still a contiguous nation, at least viewed from outside, and they still hold power throughout most of Gaul. We're a Germanic tribe seeking to be included into the foederati, and we're required to convert to Christianity. Of course, most of us don't even speak latin, and obviously we would not participate in the (very roughly speaking) inter-national diplomacy that our King is performing in order for us to move into the prosperous land of the Empire. He's the one there talking to the Romans.

It makes sense to think that leaders were, then, the first to convert, along with their closest officials; then followed by the general populace by means of authority or maybe customs (now, over the course of many years). What I'm trying to say here is that conversion to Christianity, for a Germanic tribe at least, would be a top-down affair; and would be mostly motivated by the desire to be part of the Empire, culturally and physically; hold productive land, or simply increase in power.