The main way to simplify this process is to use the services of an agent. There are a few that specialise in this and you would want to vet them a bit before putting your money, passport and credibility in their hands so I am not going to name any. But I heard about a handful that would for example arrange visas for Russia and many central Asian countries as a “package” (you provide an itinerary and your passport and they arrange invitations, send your passport to the relevant consulates for you, etc.)
As a Belgian citizen, if you do not want to go to Sub-Saharan Africa and the Middle East, this area (central Asia and the former Soviet Union) is probably the most troublesome. You will need a visa or some formalities in advance for ten or twenty other countries (India, China, North Korea, Iran, Belarus, Vietnam…) but you are lucky enough to be able to visit most of the world with little or no formalities (Europe is wide open without so much as a concern about overstaying in the Schengen area, the Americas are visa-free for you and many South-East Asian countries merely require a visa-on-arrival which is easy to get without any trip to an embassy beforehand).
The thing is is that, outside the Schengen area, visas are entirely each country's prerogative. Nobody can issue a “global” visa, it's not a mere technical issue but fundamentally at odds with the way international law and passports/visa work. One way or the other, you must obtain an authorisation to enter from each country you want to visit so having someone secure them for you one-by-one is the best you can do.
In fact, trips to various consulates are not the main difficulty with getting visas for a round-the-world journey. The main issue if you want to do it all at once and in a short time is that you often cannot get visas more than 2-3 months in advance and will often have to wait for them (at least of few days, possibly weeks) so that getting all the visas all at once before the journey might be impossible. It might also be difficult to apply from a country where you are not a resident, meaning you can't always get what you need on the road and might therefore need to return to Belgium once or twice in the middle of the trip.
Beyond visa-free travel, your location and citizenship have a few other advantages. It's (much) more expensive but Belgium offers special passports with more pages and tend to issue them quickly, which would ease the process. And with the European institutions and everything, many countries have a representation in Brussels so at least you don't have to travel very far to find their nearest consulate (according to Wikipedia, there are 185 embassies in Belgium more than in Paris, Berlin or London; by comparison Greece has 80 or so and Malta has only 20).