How do I drop all constraints from all tables?

You can derive this information easily by joining sys.tables.object_id = sys.objects.parent_object_id for those object types.

DECLARE @sql NVARCHAR(MAX);
SET @sql = N'';

SELECT @sql = @sql + N'
  ALTER TABLE ' + QUOTENAME(s.name) + N'.'
  + QUOTENAME(t.name) + N' DROP CONSTRAINT '
  + QUOTENAME(c.name) + ';'
FROM sys.objects AS c
INNER JOIN sys.tables AS t
ON c.parent_object_id = t.[object_id]
INNER JOIN sys.schemas AS s 
ON t.[schema_id] = s.[schema_id]
WHERE c.[type] IN ('D','C','F','PK','UQ')
ORDER BY c.[type];

PRINT @sql;
--EXEC sys.sp_executesql @sql;

PRINT is just there for eyeballing, not for copying and pasting the output (if you want to run it, that's what the commented-out EXEC is for) - if you have a lot of constraints, it may not show the entire script because PRINT is limited to 4,000 characters (8kb). In those cases, if you need to validate the entire script, see this tip for other ways to validate the script before running. For example:

SELECT CONVERT(xml, @sql);

Once you are happy with the output, uncomment the EXEC.

I started with the accepted answer and modified the structure to use a while loop rather than to build the full sql statement in dynamic sql. I like this better for several reasons.

The query is not stored in the large @sql variable. This implementation allows for a Print for each constraint that is dropped for logging purposes in the output. Execution seemed a little faster in my unit testing.

Set NoCount ON

Declare @schemaName varchar(200)
set @schemaName=''
Declare @constraintName varchar(200)
set @constraintName=''
Declare @tableName varchar(200)
set @tableName=''

While exists
(   
    SELECT c.name
    FROM sys.objects AS c
    INNER JOIN sys.tables AS t
    ON c.parent_object_id = t.[object_id]
    INNER JOIN sys.schemas AS s 
    ON t.[schema_id] = s.[schema_id]
    WHERE c.[type] IN ('D','C','F','PK','UQ')
    and t.[name] NOT IN ('__RefactorLog', 'sysdiagrams')
    and c.name > @constraintName
)

Begin   
    -- First get the Constraint
    SELECT 
        @constraintName=min(c.name)
    FROM sys.objects AS c
    INNER JOIN sys.tables AS t
    ON c.parent_object_id = t.[object_id]
    INNER JOIN sys.schemas AS s 
    ON t.[schema_id] = s.[schema_id]
    WHERE c.[type] IN ('D','C','F','PK','UQ')
    and t.[name] NOT IN ('__RefactorLog', 'sysdiagrams')
    and c.name > @constraintName

    -- Then select the Table and Schema associated to the current constraint
    SELECT 
        @tableName = t.name,
        @schemaName = s.name
    FROM sys.objects AS c
    INNER JOIN sys.tables AS t
    ON c.parent_object_id = t.[object_id]
    INNER JOIN sys.schemas AS s 
    ON t.[schema_id] = s.[schema_id]
    WHERE c.name = @constraintName

    -- Then Print to the output and drop the constraint
    Print 'Dropping constraint ' + @constraintName + '...'
    Exec('ALTER TABLE [' + @schemaName + N'].[' + @tableName + N'] DROP CONSTRAINT [' + @constraintName + ']')
End

Set NoCount OFF