I think you would have done it in Euro 2008, based on the fact you finished third, and there were 7 groups. So the 16 qualifiers of 2008 would have been joined in this hypothetical scenario by the 7 third-placed teams and the best fourth placed team.
However, in future when this gets implemented, they'll probably make more groups of lesser numbers of teams to avoid the scenario of four teams qualifying from one group. It'll probably be something like ten groups of five, with the winners and runners-up to be joined by the hosts and two play-off winners. So I wouldn't be taking qualification for granted. A typical group would likely contain at least one top team along the lines of England/France/Netherlands/Germany/Italy/Portugal/Spain etc. and one decent team such as one of Scandinavian countries, or Switzerland/Greece/Turkey/Ukraine or the like.
In my view this would not represent a significantly easier qualification route than the present system. Of course, this is all speculation. I have no idea what FIFA/UEFA have planned for 2016 onwards, or even whether the expansion is actuially confirmed as going ahead yet.
edit: just for fun, here's what the 2016 groups might plausibly look like based on
current FIFA Zonal rankings
A
Spain
Scotland
Finland
Austria
Moldova
B
Germany
Czech Republic
Belgium
Belarus
Georgia
C
Netherlands
Sweden
Northern Ireland
Iceland
Armenia
D
Italy
Poland
Republic of Ireland
Albania
Montenegro
E
Croatia
Ukraine
Lithuania
Norway
Estonia
F
England
Switzerland
Slovakia
Latvia
Luxembourg
G
Russia
Bulgaria
Hungary
Wales
Kazakhstan
H
Turkey
Greece
Serbia
Bosnia-Herzegovina
Azerbaijan
San Marino
I
France
Israel
FYR Macedonia
Slovenia
Malta
Andorra
J
Portugal
Romania
Denmark
Cyprus
Liechtenstein
Faroe Islands
Bearing in mind you have to finish top two to qualify (allowing for a couple of play-off spots, depending on the number of hosts), I don't think it's much different to what we have now. The main difference is that runners-up will qualify automatically rather than face a play-off, but I still assert that finishing second in any of those groups would be a pretty decent challenge.